Problem: Determine how many solutions exist for the system of equations. ${-5x+y = 7}$ ${-2x+y = -7}$
Solution: Convert both equations to slope-intercept form: ${-5x+y = 7}$ $-5x{+5x} + y = 7{+5x}$ $y = 7+5x$ ${y = 5x+7}$ ${-2x+y = -7}$ $-2x{+2x} + y = -7{+2x}$ $y = -7+2x$ ${y = 2x-7}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = 5x+7}$ ${y = 2x-7}$ The linear equations have different slopes. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ When two equations have different slopes, the lines will intersect once with one solution.